文章基本信息

标题：Complexity of the Steiner Network Problem with Respect to the Number of Terminals
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作者：Eduard Eiben ; Dusan Knop ; Fahad Panolan 等
期刊名称：LIPIcs : Leibniz International Proceedings in Informatics
电子版ISSN：1868-8969
出版年度：2019
卷号：126
页码：1-17
DOI：10.4230/LIPIcs.STACS.2019.25
出版社：Schloss Dagstuhl -- Leibniz-Zentrum fuer Informatik
摘要：In the Directed Steiner Network problem we are given an arc-weighted digraph G, a set of terminals T subseteq V(G) with |T|=q, and an (unweighted) directed request graph R with V(R)=T. Our task is to output a subgraph H subseteq G of the minimum cost such that there is a directed path from s to t in H for all st in A(R). It is known that the problem can be solved in time |V(G)|^{O(|A(R)|)} [Feldman and Ruhl, SIAM J. Comput. 2006] and cannot be solved in time |V(G)|^{o(|A(R)|)} even if G is planar, unless the Exponential-Time Hypothesis (ETH) fails [Chitnis et al., SODA 2014]. However, the reduction (and other reductions showing hardness of the problem) only shows that the problem cannot be solved in time |V(G)|^{o(q)}, unless ETH fails. Therefore, there is a significant gap in the complexity with respect to q in the exponent. We show that Directed Steiner Network is solvable in time f(q)* |V(G)|^{O(c_g * q)}, where c_g is a constant depending solely on the genus of G and f is a computable function. We complement this result by showing that there is no f(q)* |V(G)|^{o(q^2/ log q)} algorithm for any function f for the problem on general graphs, unless ETH fails.
关键词：Directed Steiner Network; Planar Graphs; Parameterized Algorithms; Bounded Genus; Exponential Time Hypothesis